Time-Resolved Particle Image Velocimetry Measurements of the 3D Single-Mode Richtmyer-Meshkov Instability

Transcription

1 Time-Resolved Particle Image Velocimetry Measurements of the 3D Single-Mode Richtmyer-Meshkov Instability Item Type text; Electronic Thesis Authors Xu, Qian Publisher The University of Arizona. Rights Copyright is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 23/08/ :18:04 Link to Item

2 1 TIME-RESOLVED PARTICLE IMAGE VELOCIMETRY MEASUREMENTS OF THE 3D SINGLE-MODE RICHTMYER-MESHKOV INSTABILITY by Qian Xu Copyright Qian Xu 2016 A Thesis Submitted to the Faculty of the DEPARTMENT OF AEROSPACE & MECHANICAL ENGINEERING In Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE WITH A MAJOR IN AEROSPACE ENGINEERING In the Graduate College THE UNIVERSITY OF ARIZONA 2016

3 2 STATEMENT BY AUTHOR This thesis titled Time-Resolved Particle Image Velocimetry Measurements of the 3D Single-Mode Richtmyer-Meshkov Instability prepared by Qian Xu has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may by granted by the head of the major department or the Dean of the Graduate College when in her or his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: Qian Xu APPROVAL BY THESIS DIRECTOR This thesis has been approved on the date shown below: 12/07/2016 Dr. Jeffrey W. Jacobs Date Department Head of Aerospace and Mechanical Engineering

4 3 ACKNOWLEDGEMENTS First of all, I would like to thank my advisor, Dr. Jeffrey Jacobs for offering me a great graduate research assistantship in his lab, and providing me with guidance and assistance through my whole graduate study and research. I would also like to thank Dr. Vitaliy Krivets for his disinterested assistance in setting up experiment, and sharing his knowledge of PIV, and Richtmyer-Meshkov instability. Working at the Experimental Fluid Mechanics & Instability lab is great fun. It is my honor to meet and work with Michael Roberts, Robert Morgan, Vladimer Tsiklashvili, Matthew Mokler, Kevin Ferguson, and Everest Sewell. Without the help and support of all of you, I could not have done my graduate study and research. Thank you. I would also thank to Dr. Steve Anderson at LaVision for his help in answer me the questions of setting up DaVis program. Finally, I would like to thank my parents. Thank you for helping and supporting me. I would like to say, you are the greatest parents in the world. I love you forever.

5 4 TABLE OF CONTENTS LIST OF FIGURES.6 LIST OF TABLES..9 ABSTRACT INTRODUCTION PREVIOUS STUDIES THEORY AND PIV TECHNIQUE Theory and Models Linear Stability Theory Zhang and Sohn Model Late-Time Models Modified 3D Sadot et al. Model The PIV Technique EXPERIMENTAL SETUP Shock Tube Driver Driven Section Test Section Speaker System Initial Perturbed Interface Laser and Optics Camera Recording System Timing Control of Experiment Experiment Timing Control DaVis Program Timing Control Gases Particle Choice, Seeding System and Size IMAGE PROCESSING, PIV VECTOR CALCULATION PARAMETER SETUP, AND MEASURMENT Image Processing PIV Vector Calculation Parameter Setup Correlation Mode Window Size, Overlap, and Iteration Initial Window Shift for Image Reconstruction (First Pass Only) Median Filter Remove Groups of Vectors, Filling Empty Spaces and Smoothing Measurement Measurement Resolution.54

6 Time Measurement Amplitude Measurement Growth Rate Measurement Circulation Measurement Uncertainty of Measurement RESULTS Results and Comparison to Linear Stability Theory Results and Comparison to Zhang-Sohn Model Results and Comparison to Modified 3D Sadot et al. Model Results and Comparison to Late-Time (Goncharov) Model Circulation CONCLUSION AND FUTURE PLAN Conclusion Future Plan 93 APPENDIX A...94 A.1. Matlab Script for Inputting Velocity and Vorticity Data A.2. Matlab Script for Circulation Calculation REFERENCE 98

7 6 LIST OF FIGURES Figure 1.1: When the shock wave passing through the perturbed interface, the misalignment of the density and pressure gradient produces the vorticity on the interface. After the shock wave passing, the piston velocity continues moving the interface downstream. The instability on the interface keeps growing, and finally turns into the structure shown above 13 Figure 1.2: Left: Laser vaporizes the cylindrical Hohlraum, which contains a spherical deuterium-tritium pellet Figure 2.1: Generation of the interface of Jones and Jacobs experiment.18 Figure 2.2: Comparison of single mode 3D Richtmyer-Meshkov instability (PLIF, Mie scattering, and numerical simulation) 20 Figure 4.1: Diagram of the shock tube, optics, and laser setup for the experiment..28 Figure 4.2: A rendering of the test section, containing front and left transparent windows for visualization, holes for generating the interface, and a hole for speaker attachment...31 Figure 4.3: A special made bottom plenum...32 Figure 4.4: A window plate for sealing the bottom plenum and letting laser travel into the test section through the bottom..32 Figure 4.5: The mounting system for the speaker. Left: the overview of the system: two aluminum plates, a speaker, four 3, ¼-20 bolts, and test section. Middle: a piece of latex sheet glued on the aluminum O-ring. Right: a steel mesh plate...33 Figure 4.6: Left is the square 3D single mode spike perturbation. And right is the square 3D single mode bubble perturbation..35 Figure 4.7. Left is the recording plane through the diagonal of the shock tube. And right is the recording plane through the saddle points 36 Figure 4.8: Side and front views of cameras and mounting system..38 Figure 4.9: Difference in percentage of velocity measurement between PIV and displacement measurement results.41 Figure 4.10: The seeding atomizer.44 Figure 4.11: PIV results of a shock passing through a single gas filled with particles.46 Figure 4.12: Mean velocity field vs. vertical distance..47 Figure 4.13: Velocity jumping part of speed vs. vertical distance, U p 1 vs. t and curve U fitting.48

8 7 Figure 5.1: Light travelling path through the corner of the test section 49 Figure 5.2: Example for image processing...51 Figure 5.3: Initial amplitude measurements of spike (saddle) and bubble (saddle) experiments...56 Figure 5.4: Initial amplitude measurement of spike (diagonal) and bubble (diagonal) experiments 57 Figure 5.5: Displacement of flat interface vs. relative time...58 Figure 5.6: The control box for the calculation of circulation...60 Figure 6.1: Experimental amplitude growth for spike (diagonal) experiment...68 Figure 6.2: Experimental amplitude growth for spike (saddle) experiment..69 Figure 6.3: Experimental amplitude growth for bubble (diagonal) experiment 70 Figure 6.4: Experimental amplitude growth for bubble (saddle) experiment 71 Figure 6.5: Comparison of experimental amplitude measurement to linear stability theory with all initial perturbation condition in non-dimensional scale...72 Figure 6.6: Comparison of experimental spike growth rate to linear stability theory in non-dimensional log-log scale...73 Figure 6.7: Comparison of experimental bubble growth rate to linear stability theory in non-dimensional log-log scale...73 Figure 6.8: Comparison of experimental amplitude measurement to Zhang&Sohn model with all initial perturbation condition in non-dimensional scale...74 Figure 6.9: Comparison of experimental spike growth rate to Zhang &Sohn model in non-dimensional log-log scale...75 Figure 6.10: Comparison of experimental bubble growth rate to Zhang &Sohn model in non-dimensional log-log scale...76 Figure 6.11: Comparison of experimental amplitude measurement to modified Sadot et al. model with all initial perturbation condition in non-dimensional scale 77 Figure 6.12: Comparison of experimental spike growth rate to modified Sadot et al. model in non-dimensional log-log scale 78 Figure 6.13: Comparison of experimental bubble growth rate to modified Sadot et al. model in non-dimensional log-log scale 78

9 8 Figure 6.14: Comparison of experimental spike growth rate to late-time (Goncharov) model in non-dimensional log-log scale 79 Figure 6.15: Comparison of experimental bubble growth rate to late-time model in nondimensional log-log scale..80 Figure 6.16: The vorticity evolution for spike (diagonal) experiment..81 Figure 6.17: The vorticity evolution for spike (saddle) experiment..82 Figure 6.18: Circulation vs. Time for spike (diagonal) 1 experiment...83 Figure 6.19: Circulation vs. Time for spike (diagonal) 2 experiment...84 Figure 6.20: Circulation vs. Time for spike (saddle) 1 experiment...84 Figure 6.21: Circulation vs. Time for spike (saddle) 2 experiment...85 Figure 6.22: Two coordinate systems, x-y for spike (diagonal) structure and x -y for spike (saddle) structure..87 Figure 6.23: Circulation vs. time for all spike experiments and comparison with Jacobs & Sheeley s circulation models.90 Figure 6.24: Sum of circulation vs. time for all spike experiments...91

10 9 LIST OF TABLES Table 4.1: List of shock tube s parts and their parameters 27 Table 4.2: Velocity measurement results of different t PIV.40 Table 5.1: Circulation results for different sizes of control box 61 Table 6.1: Experimental parameters for spike (diagonal) experiment..64 Table 6.2: Experimental parameters for spike (saddle) Experiment..65 Table 6.3: Experimental parameters for bubble (diagonal) Experiment 65 Table 6.4: Experimental parameters for bubble (saddle) Experiment...66

11 10 ABSTRACT The Richtmyer-Meshkov Instability (RMI) (Commun. Pure Appl. Math 23, , 1960; Izv. Akad. Nauk. SSSR Maekh. Zhidk. Gaza. 4, , 1969) occurs due to an impulsive acceleration acting on a perturbed interface between two fluids of different densities. In the experiments presented in this thesis, single mode 3D RMI experiments are performed. An oscillating speaker generates a single mode sinusoidal initial perturbation at an interface of two gases, air and SF6. A Mach 1.19 shock wave accelerates the interface and generates the Richtmyer-Meshkov Instability. Both gases are seeded with propylene glycol particles which are illuminated by an Nd: YLF pulsed laser. Three high-speed video cameras record image sequences of the experiment. Particle Image Velocimetry (PIV) is applied to measure the velocity field. Measurements of the amplitude for both spike and bubble are obtained, from which the growth rate is measured. For both spike and bubble experiments, amplitude and growth rate match the linear stability theory at early time, but fall into a non-linear region with amplitude measurements lying between the modified 3D Sadot et al. model (Phys. Rev. Lett. 80, , 1998) and the Zhang & Sohn model (Phys. Fluids , 1997; Z. Angew. Math Phys , 1990) at late time. Amplitude and growth rate curves are found to lie above the modified 3D Sadot et al. model and below Zhang & Sohn model for the spike experiments. Conversely, for the bubble experiments, both amplitude and growth rate curves lie above the Zhang & Sohn model, and below the modified 3D Sadot et al. model. Circulation is also calculated using the vorticity and velocity fields from the PIV measurements. The calculated circulation are approximately equal and found to grow

12 11 with time, a result that differs from the modified Jacobs and Sheeley s circulation model (Phys. Fluids 8, , 1996).

13 12 1. INTRODUCTION The Richtmyer-Meshkov instability (RMI) [1] [2] is a classical fluid instability. It is very similar to another well-known fluid instability, the Rayleigh-Taylor instability (RTI) [3] [4]. However, there is a main difference between the Richtmyer-Meshkov and the Rayleigh-Taylor instabilities. For the RMI, it occurs due to an impulsive acceleration acting on a perturbed interface between two different density fluids. For the RTI, it happens when the interface between two different density fluids experiences a constant acceleration. The RTI was initially studied by Lord Rayleigh in 1883 [3]. Then the theory was further developed by G.I. Taylor [4]. At a perturbed interface between two different density fluids, the instability occurs when the light fluid is accelerated with constant value into the heavy fluid. RTI results in the generation of vorticity that can be represented by the following formula D Dt (ω ρ ) = (ω ρ ) u + 1 ρ 3 ρ P (1.1) where ω is the vorticity vector, ρ is the density of fluid, u is the velocity vector of fluid, and P is the pressure of fluid. The last term of formula 1.1 is called the baroclinic production term that will produce vorticity at the interface when there is a misalignment of pressure and density gradients. As mentioned above, RTI only occurs when the light fluid is pushing the heavy fluid. That is When the light fluid is accelerated into the heavy fluid and ρ P < 0 (1.2) ρ P > 0 (1.3)

14 13 there is only oscillation at the perturbed interface between two different density fluids, rather than instability. The RMI was first identified by R. D. Richtmyer [1] in Further, E. E. Meshkov [2] was first to produce RMI experimentally in The RMI occurs due to an impulsive acceleration acting on the perturbed interface, rather than the RTI s constant acceleration. Furthermore, the RMI will develop no matter which fluid (heavy or light) is accelerated into the other (light or heavy). The misalignment of the pressure and density gradients will produce vorticity on the interface. Also, the transmitted and reflected shocks will generate some weaker vorticity away from the interface [5]. Figure 1.1 illustrates the production and influence of vorticity in the RMI. Figure 1.1 [5]: When the shock wave passes through the perturbed interface, the misalignment of the density and pressure gradient produces vorticity on the interface. After the shock wave passes, the piston velocity continues to translate the interface downstream. The amplitude of perturbation continues growing, finally turning into the mushroom structures shown. The growing instability distorts the interface mushroom -like structures. The heavy fluid is transported up into the light fluid to form the spikes. Conversely, the light

15 14 fluid moves downward into the heavy fluid to form the bubbles. In addition, secondary vorticity is also generated by the misalignment of the centripetal acceleration of the vortex cores and the interfacial density gradient [5]. Finally, secondary instability causes the fluid turbulent transition and the breakdown of the mushroom structures. The study of the RMI is very important to the realization of Inertial Confinement Fusion (ICF). In the United States, the largest ICF project: the National Ignition Facility (NIF) is being carried out by the Lawrence Livermore National Laboratory (LLNL). In the NIF, scientists use a spherical plastic shell filled with low-density equimolar deuterium-tritium gas as the target [6]. The D-T filled shell is placed in a small gold cylindrical container, called a Hohlraum. The Hohlraum is located at the center of a 10 meter radius spherical chamber. During the experiment, 192 lasers produce high-powered beams that illuminate the Hohlraum simultaneously. And it creates a bath of X-ray radiation that compresses and heats the D-T filled shell, until the fusion reaction occurs. Theoretically, the experiment is designed to produce more energy than the energy required for ignition. However, shock waves occur due to the force of the collapsing the sphere, resulting in the mixing of the fuel and the shell material rather than compression [6]. This phenomenon results in a negative net yield reaction, which means the produced energy is less than that required for ignition. The RMI occurs due to imperfections of the plastic shell. And it is a source of the significant mixing mentioned above. So studying the RMI is necessary for the scientists to create the compression and achieve a positive net yield reaction. Also, in astrophysics, the RMI provides some potential explanations for several findings in observation data of a supernova remnant (SNR), which are difficult to explain

16 15 in the standard model [7]. For example, the RMI can explain the formation of the radio emission and radial magnetic field in the forward shock area where the RTI fails to penetrate in. Figure 1.2: Laser vaporizes the cylindrical Hohlraum, which contains a spherical deuterium-tritium pellet. [28] What s more, the RMI plays an important role in the fuel mixing process of the supersonic combustion ramjet engines. The RMI occurs when a flame front crosses shock waves present in supersonic combustors. The presence of shock waves can be utilized to enhance the degree of mixing between fuel and oxidizer for improved combustion and engine efficiency [8].

17 16 2. PREVIOUS STUDIES In 1960, Robert D. Richtmyer [1] was first to model the RMI by applying an impulsive acceleration to the linear stability of RTI. In Richtmyer s analysis, the perturbed interface was given by: z = a cos kx ( ka 1 ) (2.1) where a is the amplitude and k is the wavenumber k = 2π λ (2.2) and λ is the wavelength. After impulsive acceleration, Richtmyer [1] found that the amplitude of initial perturbation to grow following: da dt = kva 0A (2.4) where v is the velocity of the interface right after the impulsive acceleration, a 0 is the initial amplitude of the interface immediately after the impulsive acceleration, and A is the Atwood number defined by, A = ρ (+) ρ ( ) ρ (+) +ρ ( ) (2.5) where ρ (+) is the density of heavy fluid. And ρ ( ) represents the density of light fluid.

18 17 Richtmyer [1] found that the growth rate of the interface remained constant as long as ka 1. This is also known as the linear growth regime. For larger amplitudes, ka > 1, Richtmyer suggested that the growth rate was no longer constant. In 1969, E. E. Meshkov [2] was first to experimentally validate Richtmyer s theory [1 ] of the RMI. Meshkov used a shock tube consisting of four parts: the shock tube chamber, the diaphragm, the channel, and the test section. The chamber and the channel were separated by a diaphragm consisting of four layers of cellulose acetate film, each 0.2mm thick. Before the experiment, the chamber was filled with the driven gas, nitrogen, reaching a pressure of 6.5 atm while the channel remained at atmospheric pressure. In the test section, the light gas and the heavy gas were separated by a 1- micron-thick nitorcellulose thin film. This film created a sinusoidal initial perturbation between two gases. During the experiment, the diaphragm between the chamber and the channel was broken by an exploding electric wire. Then, due to the pressure difference, a shock formed and travelled down into the test section, rupturing the thin film separating the two gases. The RMI that developed was recorded by the SFR (Spatial Frequency Response)-3M high speed camera. The experimental results obtained by Meshkov were in qualitative agreement with Richtmyer s theory [2]. However, the method of using thin film to separate the two gases has some disadvantages. The main disadvantage is that the broken film fragments will remain in the gas, and will affect the growth of the perturbation and the visualization of the flow. In addition some diagnostic techniques, such as Planar Laser Induced Fluorescence (PLIF), and Particle Image Velocimetry (PIV) are not suitable for use with the thin film method.

19 18 In 1994, another gas separation technique was developed by Brouillette and Sturtevant [9]. In their experiment, a thin (1.2 mm) horizontal metal plate was used to initially separate the two gases. During the experiment, the metal plate was withdrawn right before shock wave arriving at the interface. The resulting wake produced the initial perturbation of the interface. This technique also has some drawbacks. Firstly, it creates a relatively thick interface that will affect the growth of the perturbation. Secondly, it cannot create a perfect sinusoidal initial perturbation. Thirdly, it cannot produce the consistent initial perturbations from experiment to experiment [10]. In 1997, Jones and Jacobs [11] developed a new technique which generated a flat, membraneless interface in a vertical shock tube. In their experiment, two gases (N2 and SF6) flowed from opposite ends of the shock tube, collided and existed through slots in the test section. Figure 2.1 [11]: Generation of the interface of Jones and Jacobs experiment.

20 19 The initial sinusoidal perturbation was produced by generating a standing wave in the shock tube. The standing wave was created through oscillating the shock tube laterally. This technique successfully creates a repeatable, sinusoidal, perturbed interface without introducing any foreign material into the flow. In the Fluid Mechanics and Instability Laboratory at the University of Arizona, researchers have been studying the RMI for over 20 years. Long, et al. [12] studied the three-dimensional single mode RMI through Mie scattering, planar laser-induced fluorescence (PLIF), and numerical simulation. The experiment was carried out in a vertical mounted shock tube. The light gas was seeded with incense smoke (for Mie scattering) or acetone vapor (for PLIF) which flowed through the top end of the shock tube and the heavy gas which was unseeded, flowed through the bottom end. At the bottom of the shock tube, a bellows was oscillated by a voice coil actuator using a 4.1 Hz sinusoidal waveform that created a single-mode 3D standing wave at the interface. The RMI was generated by a weak shock wave (M=1.22) accelerating the air/sf interface. The growth of the perturbation was recorded by three high speed video cameras.

21 20 Figure 2.2 [11]: Comparison of single mode 3D RMI (PLIF, Mie scattering, and numerical simulation). Based on the experiment results, amplitude measurements from the experiments agreed well with the numerical simulation. Also, the secondary instability could be recognized along the interface from both experimental and numerical results. Aure and Jacobs performed PIV measurements of the two-dimensional single mode shock induced RMI [5]. Similar to Long et al. [12] s three-dimensional RMI experiment, Aure s experiment was also carried out in a vertical shock tube in the Fluid Mechanics and Instability Laboratory of the University of Arizona. However, the gases were seeded with spherical latex particles instead of acetone vapor or smoke and only one CCD camera was used to capture one double-frame image for each run of the experiment. A stepper motor was used to oscillate the shock tube in order to create a two-dimensional single mode initial perturbation. In this experiment, Aure and Jacobs focused on

22 21 measuring the vorticity distribution and circulation over one half wavelength and two half wavelength from velocity field measurements. Based on their results, they found that the circulation increased with time due to secondary baroclinic vorticity production. The experimental results agreed with the numerical study of Peng et al. [13].

23 22 3. THEORY AND PIV TECHNIQUE 3.1. Theory and Models Linear Stability Theory Linear stability theory was first developed for the RTI by Rayleigh and Taylor [3] [4]. Richtmyer [1] extended this analysis to the RMI by assuming the two fluids to be incompressible, irrotational, and ka<<1. In this analysis, the growth rate is predicted as d dt (a) = ka 0A V (3.1) After doing a simple integration of Eqn. 3.1, the amplitude is determined to be a = ka 0 AVt + a 0 (3.2) For a diffused interface, Brouillette and Sturtevant modified Eqn. 3.2 with a growth rate reduction factor, ψ [19] a = ka 0 AVt ψ + a 0 (3.3) where ψ is the eigenvalue of the following equation d dv (ρ dy dρ ) = dy uk2 (ρ ψ ) (3.4) Ak dy In order to solve Eqn. 3.4, an error function density profile between the two gases needs to be applied [14] y ρ(y) = ρ [1 + A erf ( δ )] (3.5)

24 23 where ρ is the average of the pre-shock and post-shock density values and δ is the maximum slope thickness. After substituting (3.5) into (3.4), one obtains d dy [(1 + Aerf (y δ )) dv dy ] = vk2 [1 + Aerf ( zy δ ) ψ k d dy erf (y δ )] (3.6) Then, by applying z = z δ, w(z) = w ( z δ ) = w (z ), and a = 1 transformed to be kδ, equation (3.6) is a 2 d dz [(1 + Aerf(z )) dw dz ] = w [1 + Aerf(z ) aψ d dz erf(z )] (3.7) Equation (3.7) is solved by splitting it into a system of ODEs. The growth rate reduction factor for SF6/Air was determined to be by Long [12] Zhang and Sohn Model Zhang and Sohn [20] [21] used a Pade approximation to construct an approximate linear solution for the RMI of incompressible fluids during the early time and introduced an asymptotic solution of the nonlinear theory for the later time. After combining both solutions, the growth rate of bubble and spike are determined by the following equations: a = a 0 1+a 0 k 2 a 0 λ 1 t+max{0,a 0 2 k 2 λ 1 2 λ 2 }k 2 a 0 2 t 2 (3.8) a bubble = a + a 0 2 kλ 3 t 1+a 0a 0 k 2 λ 4 λ 1 3 t+a 0 2 k 2 (a 2 0 k 2 λ 2 4 λ 2 3 +λ 5 λ 3 1 )t 2 (3.9) a spike = a + a 0 2 kλ 3 t 1+a 0a 0 k 2 λ 4 λ 1 3 t+a 0 2 k 2 (a 2 0 k 2 λ 2 4 λ 2 3 +λ 5 λ 3 1 )t 2 (3.10)

25 24 where λ 1, λ 2, λ 3, λ 4, and λ 5 are dimensionless functions of the post-shock Atwood number and the polar angle θ of the wave vector (kx, ky). In this three dimensional single mode experiment, the polar angle is 45 o [21]. The λ values are given by Zhang and Sohn: λ 1 = A = (3.11) λ 2 = A = (3.12) λ 3 = A A = (3.13) λ 4 = A A = (3.14) λ 5 = A A = (3.15) Late-Time Models For the late time growth rate of RMI, both Oron et al. [24] and Goncharov [25] developed similar models independently in which they found a bubble/spike = ( 1 A λ + 1) 1±A 2πt (3.16) where λ is the wavelength of the initial perturbation Modified 3D Sadot et al. Model The Sadot et al. model [22] is an empirical attempt at combining both early-time and late-time development models. For the two dimensional RMI, the model gives a bubble/spike = a 0 1+a 0kt 1+(1±A)a 0kt+ 1 2Cπ (1±A 1+A )a 0 2 k 2 t 2 (3.17)

26 25 where C = 1/3π for A 0.5 and C = 1/2π for A 0. For the three dimensional case, Chapman and Jacobs [23] modified the Sadot et al. model to agree with Zhang and Sohn three-dimensional early-time model [20] [21], and the Oron [24] three dimensional latetime model. The modified three dimensional Sadot et al. model is 1+a 0kt a bubble/spike = a 0 1+(1 λ 3 )a 0kt+ 1 (3.18) 2 (1±A)a 0 2 k 2 t The PIV Technique Particle Image Velocimetry (PIV) is a technique that visualizes the motion of particles seeded in a fluid flow and this measures the fluid velocity field. In this technique, the fluid flow is seeded with tracking particles that are illuminated by a thin laser sheet at least twice (double-pulsed) during a short time interval, dt. The images are usually recorded by high-speed digital CCD or CMOS cameras. After that, the images are divided into small interrogation windows with specific size a correlation function is applied to the interrogation windows to find the displacement of particles within each window. One of two correlation functions are typically used, the auto-correlation function and the cross-correlation function. The auto-correlation function can be applied to a double-exposed single-frame experiment. The auto-correlation function creates selfcorrelated maximum peak and two symmetric secondary peaks near the maximum one. As a result, this method requires additional information about the flow to determine which secondary peak to choose. The cross-correlation function can be applied to a double-exposed multi-frame experiment. The cross-correlation function creates only one large peak and several small peaks. The displacement of peaks is determined through the acquisition of two consecutive frames. So it does not need additional information and is

27 26 preferred in this experiment. The details of the PIV experimental setup are discussed in a later section.

28 27 4. EXPERIMENTAL SETUP 4.1. Shock Tube The experiments described here have been carried out using the shock tube in the Fluid Mechanics and Instability Laboratory at the University of Arizona. The shock tube consists of three main parts: the driver, the driven section, and the test section. The driver is the same one that Morgan [14] and Aure [15] used for their MS research projects on the RMI. The driven section is made up of a long section which was used on the research projects of Morgan and Aure and a new short section which was manufactured in the machine shop at the University of Arizona. The test section is the same one that Morgan used for his PhD research project on the rarefaction wave driven Rayleigh-Taylor instability [16]. Shock Tube Parts Length (m) Cross Section Dimensions (mm) Material Driver (radius of circle) Fiberglass Old Driven Section x88.9 (square) Fiberglass New Driven Section x88.9 (square) Acrylic, Metal Test Section x88.9 (square) Acrylic, Metal Table 4.1: List of shock tube s components and their dimensions

29 Figure 4.1: Diagram of the shock tube, optics, and laser setup for the experiment. 28

30 Driver The Driver section is a cylindrical pipe that is made of fiberglass. The length of the section is 3.67m and the inner diameter is 101.6mm. The fiberglass driver section can hold the pressure up to 150 psi, which is high enough to create the required shock speed of the experiment. A solid fiberglass flange seals the top end of the driver section. The solid fiberglass flange also has two pipe fittings that connect to the external nitrogen gas tank. Also, counter weights are connected to the top of the driver section through a pulley system. The counter weights are designed to lift the driver section easily for cleaning the driven, test sections, and replacing the polypropylene sheets. The bottom end the driver section is attached with a special made plenum Driven Section The total length of the driven section is 4.96m. It is made of fiberglass with hollow square cross section. The dimensions of inner cross section are 88.9x88.9mm. The top end of the driven section is connected to a special made plenum. The plenum has thread on the outside surface in order to connect the driver and driven sections through screwing one part on the other. Additionally, the solenoid is mounted inside the top plenum. What s more, the top plenum has two pipe fittings that connect to the external light gas (air) tank. The light gas (air) enters the driven section at 6L/min. It is similar to the one that Ferguson used for his MS thesis research [10]. During the experiment, the solenoid is fired to break two 23μm thick polypropylene sheets that are used for the separation of the driver and driven sections. After breaking polypropylene sheets, the shock is formed due to the pressure difference between the driver and driven sections.

31 30 The driven section includes two parts, a 4.09m long section and a 0.9m short section. Two pressure transducers are mounted on the side wall of the short driven section. The distance between them is 600mm. And the lower pressure transducer is located 140mm above the bottom of the driven section. During the experiment, the pressure transducers are used to determine the shock speed. The pressure transducers are connected to an Agilent 53131A Universal Counter to record the shock s transit time between them. The shock speed can be calculated through V shock = L t (4.1) where L is the distance between two pressure transducers and t is the time recorded on the counter. A regular flange is attached at the bottom end of the driven section to connect to the test section Test Section The test section has a flange at its upper end that allows it to be connected to the driven section. The test section is 0.92m long and has an 88.9mm X 88.9mm square inner cross section. The front and left walls are made from 1.1cm thick clear acrylic. The back and right walls are made from 1.1cm thick black acrylic plate. Instead of using bolts to assemble the section, all four walls are glued carefully together in order to record clear images through the diagonal direction. On each side of the walls, there are six 2.3mm diameters holes. These holes are 18 cm below the top end of the test section. Near the bottom of the test section, there is a hole with 7.5cm diameter. The center of the hole is 9cm above the bottom of the test section. The hole is designed for attaching the speaker to create the initial perturbation. The drawing of the test section is shown in Figure 4.2.

32 31 Figure 4.2: A rendering of the test section, containing front and left transparent windows for visualization, holes for generating the interface, and a hole for speaker attachment. The bottom of the test section is attached to another special made plenum that allows the heavy gas (SF6) to enter the test section at a rate of 6L/min. The light and heavy gases enter the test section from the opposite direction, and exit through the 2.3mm holes on the sidewalls. As a result, a flat initial interface is produced.

33 32 Figure 4.3: A special made bottom plenum. A window plate is designed to seal the open end at the bottom of the test section. The center of the plate is a clear 8 cm diameter circular glass window that allows the laser to enter and form a thin sheet for visualization in the test section. Figure 4.4: A window plate for sealing the bottom plenum and letting laser travel into the test section through the bottom.

34 Speaker System A Tang Band W5-1138SM 5-1/4 Neodymiun Subwoofer speaker is mounted at the bottom of the test section. Due to the shock passing during the experiment, there is an impulsive pressure jump in the test section. It was therefore necessary to build a solid mounting system, in order to protect the speaker from the pressure jump. The mounting system of the speaker was designed and manufactured by Matt Mokler and Justine Schluntz. In this system, a piece of 1.5cm thick aluminum plate is bolted at the bottom hole of the test section through six 8-32 screws. Also, a groove was manufactured in the plate for placing a piece of thin latex sheet. The latex sheet is glued on an aluminum O- ring, and backed by a steel mesh plate that is designed to prevent the latex sheet from breaking and protecting the speaker. The speaker is mounted to another piece of 1.5cm thick aluminum plate. The speaker is attached to the test section by connecting the two aluminum plates using four 3, 1/4-20 bolts. Figure 4.5: The mounting system for the speaker. Left: the overview of the system: two aluminum plates, a speaker, four 3, ¼-20 bolts, and test section. Middle: a piece of latex sheet glued on the aluminum O-ring. Right: a steel mesh plate. The 3D single mode initial perturbation can be created by oscillating the gas vertically in the test section. The oscillation can be generated by the oscillation of the

35 34 speaker system. During the experiment, a Krohn-Hite Corp., model 2400 function generator is used to generate a sinusoidal signal at a specific frequency. The signal is amplified by a Labworks Inc., Pa-119 power amplifier and then sent to the speaker. The speaker drives the latex sheet at the same frequency and amplitude to oscillate the gas in the test section Initial Perturbed Interface As mentioned above, the 3D single mode initial perturbation is generated through oscillating the gas by the speaker. The initial perturbation can be modeled by the following equation: η(x, y ) = a(t) cos (k 1 x )cos (k 2 y ) (4.2) where η represents the position of the interface at given time, x and y values; a represents the amplitude of the sinusoidal wave; x and y represent the horizontal coordinates in diagonal directions of the test section and k 1 = k 2 = 2π λ (4.3) where λ is the wavelength which in this case is equal to the diagonal length of the test section. λ = 2w (4.4) where w is the width of the test section. The wave number, k of the 3D single mode initial perturbation can be determined by:

36 35 k = k k 2 2 = 2π w (4.5) and the frequency of the periodic speaker oscillation is determined by 1 f = 2π ω = 2π Agk (4.6) where A is the Atwood number (see Eqn. 2.5); g is the gravitational acceleration. In this experiment, the Atwood number of air/sf6 is approximately 0.63 as determined by Robert Morgan in his PhD research [16]. The theoretical frequency of the signal is calculated using (4.6) to be 3.32Hz. The actual frequency that is used for driving the speaker is 3.24Hz (0.08Hz less than the theoretical value). The initial 3D single mode perturbation modeled by MATLAB is shown below: Figure 4.6: Left is the square 3D single mode spike perturbation. And right is the square 3D single mode bubble perturbation. For each initial 3D single perturbation, images are recorded in two different planes, one through the diagonal of the shock tube, and one through the saddle points,

37 36 Figure 4.7. Left is the recording plane through the diagonal of the shock tube. And right is the recording plane through the saddle points Laser and Optics In order to provide sufficient light for recording the PIV images, a 75W Photonics 527nm Nd: YLF pulsed laser is used in the experiment. The laser has a maximum output of 55mJ energy per pulse. The frequency of the laser pulses is controlled externally by the LaVision DaVis software package, which is simultaneously adjusted to the frequency of the recording cameras. For the double-pulse PIV experiment, there are two separated laser pulses fired in quick succession. The time between two laser pulses is also controlled externally by the DaVis program. The diode current is set to be 35.5A for a PIV experiment using air and SF6. For a successful PIV experiment, it is necessary to produce approximately the same energy output for each pulse. At the same time, the quality of the laser sheet is critical to obtaining good results [15]. The laser is carefully

38 37 levelled on an optical table. After the laser beam leaves the aperture, it is first reflected by a 527nm coated mirror that is oriented in 45 o and mounted at the edge of the optical table. This mirror reflects the laser beam 90 o, turning it from horizontal to vertical. Next, the laser beam is reflected by another 527nm coated mirror that is mounted on an optical trail on the ground. This mirror turns the laser beam from vertical to horizontal, where it then reflects from a third 527nm coated mirror mounted immediately below the test section. After being reflected by the third mirror, the laser beam passes through a CVI PLCX C-527, f = +1000mm spherical lens and a CVI RCC C-527, f = -30mm cylindrical lens, producing a thin laser sheet passing upward in the test section for particle illumination. For different cameras views of initial perturbations, i.e., spike (saddle), bubble (saddle), spike (diagonal), and bubble (diagonal), the distance between the spherical and the cylindrical lens is varied Camera Recording System In the experiment, Photron FastCam APX RS camera are used. The cameras have 10-bit CMOS (Bayer system color, single sensor) sensors with 17μm pixels. In addition, the cameras have a 16.7ms to 2μs global electronic shutter independent of frame rate and overexposure protection [17]. The maximum camera resolution is 1024 x And the highest frame rate (fps) is 250,000 for reduced resolution. During the experiment, three cameras are used with resolution 1024 x 640 and at frame rate (fps) of 2000Hz. The cameras are bolted together vertically using two acrylic bars. Both of the acrylic bars are machined with slots to move each camera freely in vertical direction. The distance between cameras is adjustable by inserting acrylic blocks with different thicknesses. The three-camera assembly is clamped using two large right-angle brackets, and is mounted

39 38 on a vertical camera-mounting rail that is parallel to the test section. The cameramounting rail allows the whole three-camera assembly to be adjusted freely to cover different sections of the test section in vertical direction. For each camera, a 50mm focal length Nikon lens with aperture 1.2 is chosen to focus the particles in the laser sheet plane. Figure 4.8: Side and front views of cameras and mounting system Timing Control of Experiment The timing control system of the experiment is divided into two parts: the experiment timing control and DaVis program timing control. The experiment timing control includes monitoring the pressure in the driven section, firing the solenoid at the right time, triggering data acquisition when shock passes, setting the phase delay for the initial perturbation, and sending the signal to the DaVis program. The DaVis program timing control focuses on controlling the time delay between laser and cameras triggers, setting a time delay ( t PIV ) for the double-pulsed laser and setting the frequency and exposure time for the cameras.

40 Experiment Timing Control In the experiment timing control system, an Arduino program was written to control the firing time of the solenoid. In this program, the time of firing of the solenoid is determined by two criteria, the pressing of a button and waiting the peak of the sinusoidal signal from the function generator. A MKS 250 controller monitors the pressure of the driven section. Immediately before the pressure of the driven section reaches 35.5psi, the function generator, amplifier and speaker are turned on. The function generator continues sending the sinusoidal signal to the Arduino. At the time of pressure reaches 35.5psi, the button is manually pressed and the Arduino program waits for the peak of sinusoidal signal from the function generator. When both criteria are met, the solenoid fires and punctures the polypropylene sheets. A shock wave then forms and travels down in the shock tube. When the shock wave reaches the pressure transducer that is closest to the interface. It sends a signal to channel A of a Stanford Research System DG 535 digital delay/pulse generator. The trigger is set to be external with Threshold: 0.2V, Slope: positive (+), and Trigger Term = HighZ. The delay of channel A is 0 second. The output is set to be Load: 50 Ω, TTL, and Signal: Normal. Finally, the digital delay/pulse generator sends a TTL trigger signal to the LaVision Highspeed Controller that starts the DaVis program DaVis Program Timing Control In the DaVis program, there are several timing controls. The first is the reference time which is the time between the external trigger input and image acquisition [18]. In the experiment, the reference time is set to be 0ms. The second timing control is the delay

41 40 time for each laser pulse, T1A and T1B. These are also set to be 0ms. The third control is the time delay ( t PIV ) between the two laser pulses. This time delay is critical for obtaining the desired displacement of the tracer particles and thus the quality of the fluid velocity measurements [15]. Ideally, the optimal t PIV should be equal to the time required for each particle to be displaced by 1-2 particle diameters. And it is also related to the correlation window size. Finding this optimum can be difficult in some fluid experiments. However, the Richtmyer-Meshkov experiment provides a convenient way to optimize ( t PIV ) experimentally. In this process an experiment is run with a shock passing through a flat interface and then measuring the velocity of interface using two methods: directly using the displacement based on measuring the interface location in two images and indirectly by using PIV. In this process, both top and bottom gases are air and only the top gas is seeded with particles in order to visualize the interface clearly. The direct and PIV velocity measurement for t PIV = 5μs, 10 μs, 12 μs, 15 μs, 17 μs, 20 μs are listed in Table 4.2 and Figure 4.8 shows a plot of percent error as a function of dt. dt (us) PIV (m/s) Interface (m/s) % Difference Table 4.2: Velocity measurement results of different t PIV

42 % Error 41 % Difference vs. dt(us) dt(us) Figure 4.9: Difference in percentage of velocity measurement between PIV and displacement measurement results. Based on Table 4.2 and Figure 4.9, the % difference decreases rapidly by increasing t PIV and t PIV =15 μs gives the minimum % Difference between PIV and displacement measurement results. So for the 3D single mode RMI PIV experiment, t PIV is set to be 15 μs. And the particles are moving around 5 pixels in t PIV. The last timing control in the Davis program is the exposure time of the camera. Because of the double-pulse PIV setup, the exposure time is automatically set by DaVis based on t PIV and the recording frequency. Instead of the timing controls mentioned in and 4.6.2, there are some other critical timing controls in the experimental setup. It is desirable to have the perturbation amplitude to be either maximum or minimum to produce either a spike experiment or a bubble experiment. However, due to the shock s travelling time in the tube and the phase delay between the speaker oscillation and the interface perturbation, the center of interface perturbation may not be spike or bubble at the time of shock arrival. So it is

43 42 necessary to add a delay time before firing the solenoid. The time is determined by the following procedure. The procedure is divided into three parts. In the first part, a recording is made of the interface oscillation using a high-speed camera operated at a very high frame rate (f). The recording time should contain at least one oscillation period of the interface movement. In the second part a RMI shock tube experiment is run without a time delay. From this experiment, an image of the interface is extracted when the shock is just arriving at or is very close to the interface. The third part of the procedure then determines the time delay. A frame from the movie of the first part is found which is similar or identical to the image of the interface of the second part. Then the number of frames (N) between this frame and one where the desired bubble or spike perturbation is observed is found. The delay time is approximately equal to t delay = N f (4.7) In section 4.6.1, it is mentioned that the bottom pressure transducer sends a signal to a digital delay/pulse generator to trigger the start of the experiment. At the same time, both pressure transducers are connected to an Agilent 53131A Timer/Counter. The timer/counter records the transit time of the shock wave travelling between the two transducers. Knowing the distance between two transducers, the shock speed can be easily calculated. It is very important to maintain the consistency of the shock speed from experiment to experiment Gases In the experiment, Nitrogen is used for pressurizing the driver section. Air and SF6 are used as light/heavy gas pair in the test section. The Atwood number for this gas

44 43 combination is On a manifold panel, several flowmeters monitor and control the flow rates of all the gases Particle Choice, Seeding System and Size In order to successfully implement PIV in the experiments both gases must be seeded with tracking particles. On the one hand, the seeding particles must be small and light enough to move with the natural flow accelerations, so that the measured velocity field is the same as or close to the real velocity field. If the size of particles is too large or the density of particles is too high, the particles motion will have a significant lag in response to the real flow movement. On the other hand, in order to produce experimental images with high signal to noise ratio, the seeding particles must be large enough to scatter sufficient light from the laser illumination. Also, the choice of the particles is limited to the other factors, such as the temperature of the flow, the cleanliness requirement [15], and the chemical and physical reactions between particles and seeded flow. When it comes to the choice of particles, all these factors should be considered. Based on the previous PIV experiments, several kinds of particles were considered for Air/SF6 RMI experiments in the shock tube. The simplest and cleanest particles are water droplets. The size of water droplets varies from 0.1μm to 0.95μm. However, the problem with using water droplets is that they evaporate very quickly when high intensity laser light illuminates them. Solid particles, such as TiO2 were also considered. TiO2 are usually used for the visualization of flows under extreme conditions, such as high temperature and pressure. The size of TiO2 particles varies from 0.1μm to 0.3μm. However, the problem with using solid particles, such as TiO2 is that the particles

45 44 can scratch and cover the inside of the shock tube. Additionally, Roger Aure used polystyrene latex spheres as particles in his PIV measurements of the single mode shock induced RMI [15]. The sizes of particles used by Aure were 0.09μm and 0.3μm. They performed very well with the high energy double pulsed Nd:YAG laser used in these experiments. However, in the present time resolved experiments that require the use of a lower energy laser and less sensitive cameras, the light output using these particles is too small to produce good results, yielding much lower signal to noise ratio of the recorded images. In the experiments, an atomizer was designed to generate the seeding particles through the vaporization of a 50/50 mixture of propylene glycol and glycerin. The seeding atomizer is shown below: Figure 4.10: The seeding atomizer. The liquid reservoir is used to contain the 50/50 propylene glycol and glycerin mixture. A metal coil with a silica wick is placed in the reservoir. The silica wick is soaked with the

46 45 liquid mixture. An external power supply connected to the coil controls the current applied to the coil. In the experiments two atomizers were used. The one for air was operated at 0.68A and the one for SF6 was operated at 1.06A. Different currents are necessary because the seeding density was found to be a function of coil current and gas properties. Thus the current was adjusted in order to produce a very slight difference seeding density in the two flows in order to help visualize the interface in the experiments. During an experiment, the gas flows into the reservoir from the top. The electrified coil then heats and vaporizes the propylene glycol and glycerin liquid mixture on the silica wick. Glycol particles are generated and which then mix with the gas in the reservoir. The seeded gas then exits through the side hole of the reservoir, and flows through a desiccant dryer to remove any excess vapor before entering the shock tube. Two independent seeding systems continuously generate propylene glycol particles for both the top (air) and bottom (SF6) gases with the proper density. During the experiment, the particles are illuminated by the laser sheet and provide enough light intensity for the flow visualization. In order to estimate the size of the propylene glycol particles, Stokes approximation is applied. If the density of the particles is much larger than that of the fluid, the step response time of the particles satisfies the following exponential equation: U p = U [1 exp ( t τ s )] (4.8) where Up is the velocity of the particles, U is the post shock gas (or piston) velocity, and τ s is the relaxation time given by

47 46 τ s = d p 2 ρ p 18μ gas (4.9) where dp is the diameter of the particle, ρ p is the density of the particle and μgas is the dynamic viscosity of the gas. In order to obtain an estimate of the particle size used in these experiments, PIV measurements were performed to obtain the velocity field produced by the passage of a shock wave through a single gas. Figure 4.11 shows the velocity measurements from one of these experiments. Figure 4.11: PIV results of a shock passing through a single gas filled with particles. After averaging the velocity vectors in the horizontal direction, the mean velocity field along the vertical distance is obtained and shown in Figure 4.12.

48 47 Figure 4.12: Mean velocity field vs. vertical distance. Then applying a Galilean transformation to the V vs Y plot by dividing the y position by the shock speed provides a plot of the mean particle velocity field vs. time as is shown in Figure At the same time, Eqn. 4.8 can be transformed into U p U 1 = exp ( t τ s ) (4.10) allowing the relaxation time to be easily determined by fitting a simple exponential function to the rapidly changing part of the U p U 1 vs. t plot.

49 48 Figure 4.13: Velocity jumping part of speed vs. vertical distance, U p U fitting. 1 vs. t and curve Figure 4.13 shows that the relaxation time is determined to be μs thus making the diameter of the particles to be 2.35μm.

50 49 5. IMAGE PROCESSING, PIV VECTOR CALCULATION PARAMETER SETUP, AND MEASUREMENT 5.1. Image Processing During the experiment, raw image sequences are acquired by the cameras. It is necessary to process these images in order to reduce noise and hence, to increase the signal to noise ratio. In some experiments, the cameras view the instability through the corner of the test section. Thus, the corner is also recorded in the raw images. In order to remove this artifact, a vertical rectangular slice encompassing the corner is removed from the raw images. As mentioned in Long s thesis [12], it is guaranteed not lose any information as the light travels through the corner of the shock tube as shown in Figure 5.1: Figure 5.1: Light travelling path through the corner of the test section. Another artifact that must be dealt with results from the fact that laser also illuminates the side wall of the test section producing bright oversaturated regions in the images. Thus, portions of the illuminated walls are also removed from the raw images in

51 50 order to reduce these artifacts. All these editing processes are performed using the DaVis built in define geometric mask function. Next, the raw images also passed through the image post-processing function in the DaVis program. The image post-processing function contains processes such as inversion, subtract sliding background, subtract offset, and particle intensity normalization. The inversion process is very helpful when the particle intensity is lower than that of the background. In this experiment, the particle intensity is always brighter than the background, thus this function is disabled during the image processing. The subtract a sliding background function works as a high pass filter which filters out the large intensity fluctuations in the background due to reflection and other similar artifacts, while keeps the small intensity fluctuations [26]. This function helps to maintain a constant background intensity level. However, the main areas of intensity fluctuations are located at the corners and the side walls. Since these areas have already been removed as described above, this function is also disabled. The subtract an offset function is useful for correlation function normalization and the offset is set to be specified in counts (intensity) [26]. And finally, the particle intensity normalization function is very helpful for correcting the particle intensity in a local window with defined scale length. This function determines the local maximum and local minimum intensity in that window, and calculates the median intensity. After that, the original particle intensity is normalized to the median intensity. This process produces homogenous particle intensities and enhances the signal to noise ratio of the images.

52 51 Original Particle Image After Image Processing Particle Image Figure 5.2: Example of the results of image processing PIV Vector Calculation Parameter Setup After processing the raw images, it is time to calculate the velocity field using a PIV algorithm. There are many commercial PIV programs available on the market as well as open source programs freely available. In this study, the DaVis package from LaVison is used for the PIV calculations. Before running the program, some vector calculation parameters need to be set up Correlation Mode For PIV vector calculation, images are divided into so-called interrogation windows. In one interrogation window, the velocities of all particles are assumed to be the same. The correlation function processes the intensities in the interrogation window and determines one velocity vector for each interrogation window. The correlation function used here is the sequential cross-correlation function which applies crosscorrelation to each experimental double-frame image pair and calculates the vector field from them [26].

53 Window Size, Overlap, and Iteration The window size parameter defines the interrogation window size. In the DaVis program, the window is set to be square with a dimension of 16, 32, 64, 128, 256, 512 or 1024 pixels. In this experiment, the entire interrogation process is divided into two steps. The window size is set to be a multi-pass interrogation window with decreasing size from to pixels. For both sizes of interrogation windows, the weighting function factor is set to be 1:1. A Gaussian weighting factor of 1:1 is more accurate in calculating the velocity field. But it will take more time for computation. The overlap parameter defines the overlap between two interrogation windows next to each other. The larger the overlap, the larger the density of vectors calculated in the result. For example, with no overlap a pixel image yields approximately 320 vectors when choosing an interrogation window size of pixels. If the overlap is increased to be 50% with the same window size, there will be approximately 1209 vectors in the result. In this study, using a multi-pass interrogation window, the overlap of the first pass window with pixels is 50% and the overlap of the second pass window with pixels is 75%. For the multi-pass with decreasing window size operation, the number of iterations can also be selected in the DaVis program. For this study the number of iterations is set to 1 for the first pass, and 2 for the second pass Initial Window Shift for Image Reconstruction (First Pass Only) In this function, an estimated correlation peak displacement can be entered into program before correlation. A good estimation will improve the signal-to-noise ratio of

54 53 the correct correlation peak [26]. In this study, the initial window shift is set to be constant and equal to the piston velocity in the y-direction Median Filter For both multi-pass post-processing and vector post-processing, the median filter built into the DaVis program is useful for removing and replacing spurious vectors. The median value at a vector location is calculated using its eight neighbor vectors. The criteria for removing or keeping a vector with velocity components U and V is [26] U median U rms U U median + U rms And (5.1) V median V rms V V median + V rms where median represents the median value of all neighboring vectors. And rms represents the root mean squared deviation of all neighboring vectors. In DaVis, two median filters can be selected, local median filter, remove & replace and regional median filter, strongly remove & iteratively replace. For local median filter, remove & replace, the basic algorithm is shown in Eqn For regional median filter, remove & replace, a four-pass filter is applied. The first pass filters out vectors that do not satisfy the criteria in Eqn The second pass removes vectors that do not have a sufficient number of neighboring vectors. The third pass fills in empty vector location with good vectors as described below. The last pass throws out groups of vectors with good vectors less than a certain number.

55 Remove Groups of Vectors, Filling Empty Spaces and Smoothing Sometimes, it is possible to obtain incorrect groups of vectors due to an error in the image, such as wall reflection [26]. At the same time, these vectors might have similarly incorrect vectors in their neighborhood. So the local median filter strong remove & iteratively replace cannot filter them out. The remove groups function helps the local median filter strong remove & iteratively replace operation increase the confidence of the measurements. It removes the vector groups with less than N confident vectors [26]. After removing the bad vectors, the empty space can be filled with an average of all non-zero vectors in the neighborhood. The noise of the final vector field can be reduced through a smoothing function. This method is very useful for the calculations of vorticity Measurement Measurement Resolution For the PIV measurements, the overall size of the experimental image is 1024 x 640 pixels (228mm x 144 mm) and the pixel resolution is 0.225mm. Also, (128 x 80) velocity vectors are determined for each double-pulsed image pair. The distance between individual vectors is 8 pixels (or 1.55mm). Once the flow velocity fields are calculated, it is time to obtain measurements of several important parameters for the further data analysis and model comparison. These parameters are: time, amplitude, growth rate, and circulation.

56 Time Measurement In the experiment, cameras begin acquiring images when the shock wave passes the pressure transducer located above the initial interface position. It is necessary to determine the relative time, t0, corresponding to when the shock arrives at the initial interface. The method of measuring the exact time is discussed below. First, the first image acquired after the shock passes through the initial interface is called Image 1. Because the camera recording frequency is 2000 image pairs per second, the time between each image pair is fixed to be 1/2000s. If the relative time, t1= t t0, of image 1 is determined, the relative time of the following images (Image 2, 3, 4..) is easily determined by adding 1/2000s to the time of the previous image. Due to high image acquisition speed of the cameras, the shock wave is captured in image 1. The distance, d0_1, between the initial interface and shock wave is measured directly from this image. The next step is the measurement of the speed of the shock wave. Because the shock wave can be observed in both Images 1 and 2, the distance, d1_2, that shock wave travels can be measured from these images. At the same time, the time between Image 1 and 2 is fixed to be 1/2000s. The shock wave speed is then calculated using V shock = d 1_2 t 2 t 1 (5.2) The relative time, t 1, of Image 1 is then t 1 = d 0_1 V shock (5.3) The relative time, t i, of rest images is

57 56 t i = t (i 1) (5.4) 2000 where i represents the image number Amplitude Measurement The initial amplitude plays an important role in modeling RMI. Because the experiment is three-dimensional and single-mode, for the spike (saddle) and bubble (saddle) experiments, the initial amplitude is the vertical distance between the tip of the initial spike/bubble structure and the interface edge (saddle point) at the wall of shock tube. Y 0 Y X Initial Amplitude Initial Amplitude 0 X Figure 5.3: Initial amplitude measurements of spike (saddle) and bubble (saddle) experiments. For spike (diagonal) and bubble (diagonal) experiments, two different methods of amplitude measurement are introduced. One is similar to as described above in which initial amplitude is the half total vertical distance between the tip of the initial spike/bubble structure and the interface edge in the corners of shock tube. The other method attempts to find the saddle point locations along the interface. Assuming the initial perturbation is a perfect sine curve, the horizontal, X location of saddle point is X saddle = X edge+x tip 2 (5.5)

58 57 The amplitude is then determined through measuring the vertical distance between the tip of initial spike/bubble structure and saddle point locations. Y 0 Y Saddle Point X Initial Amplitude Initial Amplitude 0 Saddle Point X Figure 5.4: Initial amplitude measurement of spike (diagonal) and bubble (diagonal) experiments. After measuring the initial amplitude, the next step is to determine the amplitude of the growing spike or bubble after shock passage through the interface. Due to wall effects, some small interface instabilities are observed to develop near the wall that make it difficult to directly measure the Y value there, especially during the later times of the experiment. In order to solve this problem, the amplitude is determined through measuring the distance between spike/bubble location and the location of the interface of an equivalent flat interface experiment at the same time. Assuming the location of the initial flat interface is the same as Y=0 in the perturbed experiment, the Y value of spike/bubble, Y spike/bubble, is directly measured in each image. The Y value of flat interface is calculated using Y flat interface = V piston t i (5.6) where Vpiston is the piston velocity or the flat interface velocity found in an experiment using the same gases and shock Mach number and ti is the relative time defined in

59 Displacement (mm) 58 After running a flat interface (SF6/Air) experiment with the same Mach number (1.19) as the perturbed RMI experiment, the flat interface piston velocity is determined by finding the slope of the measured displacement of the flat interface versus time. The displacement of flat interface is measured relative to the initial interface positon and the relative time is calculated using the method described in section Displacement vs. Time y = 66185t R²= (Flat Interface Exp 1) y = 66002t R²= (Flat Interface Exp 2) y = t R²= (Flat Interface Exp 3) FlatInterface 1 Flat Interface 2 Flat Interface Time (s) Figure 5.5: Displacement of flat interface vs. relative time. Figure 5.5 shows measured displacements for three experiments from which flat interface piston velocities are calculated to be m/s, m/s, and m/s. The average of these values is m/s which was used in the amplitude measurements. The amplitude of spike /bubble at time, t is then equal to A spike / bubble = Y spike/bubble Y flat interface (5.7)

60 Growth Rate Measurement measurements In these experiments the growth rate, a, can be directly determined from the PIV a = V spike/bubble V piston (5.8) where Vspike/bubble is the measured vertical velocity at the tip of spike or bubble. Also, the initial growth rate, a 0, can be calculated using the result from linear stability analysis a 0 = a 0 V piston ka (5.9) where a 0 is the initial amplitude, V piston is the piston velocity, k is the wave number, and A is the Atwood number Circulation Measurement Because both the velocity field and the vorticity field can be obtained from the PIV measurements and since this is a vortex dominated flow, it is instructive to consider the measurement of circulation in one half wavelength of the instability. By definition, circulation can be obtained by either a line integral around a closed curve of velocity field or an area integral on a closed area of vorticity field. Also, only velocity components in x and y directions are determined in this 3D single mode experiment. So a circulation measurement is taken based on the following two methods: Γ = ω z da (5.10) Γ = udx + vdy (5.11)

61 60 Both the line integral of the velocity field and the area integral of the vorticity are applied to estimate the circulation. Similar to Aure s method [15], a rectangular control box is used to define the area of integration. An example of a typical control box is shown in Figure 5.6: Figure 5.6: The control box for the calculation of circulation. As shown in Figure 5.6, the entire vortex structure is enclosed in the control box. One major concern is whether changing the size of control box will change the estimation of circulation. In order to determine this effect, different sizes of control boxes were used for the calculation of circulation for the same experimental image. The result of this study is that changing the size of control box produced little effect on the estimation of

62 61 circulation, as long the main vortex structure is kept inside the control box. This indicates that it is not necessary to keep a constant-sized control box for all experimental images. Table 5.1 contains the circulation results for different sizes of control box in Figure 5.6. Size (Pixels) Circulation (m 2 /s) Average (m 2 /s) 365x x x x Standard Deviation (m 2 /s) 365x x x Table 5.1: Circulation results for different sizes of control box. For the line integral of the velocity field, both forward Euler and trapezoidal methods were applied and for the area integral of vorticity, the trapezoidal method was used. The MATLAB code for the calculation of circulation is contained in the appendix Uncertainty of Measurement In the experiments, uncertainties have three major sources, error in the PIV measurements based on the PIV algorithm implemented in the DaVis program, uncertainty (or lack of repeatability) of the initial perturbation, and uncertainty of the amplitude measurement. One way of estimating the accuracy of PIV measurements is to assume that the PIV algorithm can measure the displacement of particles within a measurement volume to within 0.05 pixels. Using this assumption along with the pixel

63 62 resolution of mm and dt of 15μs results an error of 0.742m/s in velocity measurement. At the same time, the velocity uncertainty of each experiment can also be calculated by the uncertainty function which is built in the DaVis program. This built-in function measures the velocity uncertainty from correlation statistics that investigate the intensity differences of individual particles and their contributions to the shape of correlation function, and then take the standard deviation [29]. For the entire velocity field, the average of the uncertainty calculated by the DaVis program were found to be 0.590m/s for the spike (diagonal) experiments, 0.554m/s for the bubble (diagonal) experiments, 0.567m/s for the spike (saddle) experiments, and 0.581m/s for the bubble (saddle) experiments. Thus the uncertainty function built into DaVis gives a smaller, but similar in size, velocity error than that from sub-pixel accuracy calculation. As mentioned above, another source of uncertainty is the lack of repeatability of the initial perturbation. Theoretically, the initial perturbation is supposed to consist of a square 3D single mode. However, due to the exit holes in the shock tube and viscous effects at the corners and walls, it is impossible to form an interface in the precise singlemode as expected and errors are introduced into the experimental measurements. For example, it is very difficult to accurately measure the amplitude of initial perturbation, because the interface near the wall is not perfectly flat. The last source of uncertainty is the amplitude measurement. In the experiments, the spike/bubble amplitude is measured based on Eqn. 5.7: A spike / bubble = Y spike/bubble Y flat interface (5.7) where the flat interface displacement is determined by piston velocity

64 63 A spike / bubble = Y spike/bubble V piston t (5.12) Again, using the linearized approximation, the uncertainty of the spike/bubble amplitude is: A ΔA s/b = ±[( ΔY Y spike/bubble ) 2 A + ( ΔV spike/bubble V piston ) 2 + ( A piston t Δt)2 ] 1/2 (5.13) where ΔY spike/bubble is the uncertainty in physical measurement, which is approximately ± one pixel (± 0.225mm). ΔV piston is the average piston velocity uncertainty that is determined to be 0.588m/s. And Δt is the uncertainty in time measurement, which is equal to 0s. Then, the uncertainty of spike/bubble amplitude can be obtained as a function of time: 2 ΔA s/b = ±[(ΔY spike ) + (tδv piston ) 2 ] 1 2 = ±[ (588 t) 2 ] 1/2 mm (5.14) bubble

65 64 6. RESULTS In the experiment, four initial perturbation conditions are studied: Spike (diagonal), Bubble (diagonal), Spike (saddle), and Bubble (saddle). For all four conditions, the wavenumbers are the same equaling (1/mm). The amplitude, growth rate, and circulation are calculated and compared with different models. The experimental parameters for each initial perturbation condition are listed in Tables 6.1 through 6.4. Spike (diagonal) Parameter Symbol Value Initial Amplitude (mm) a Wave Number (1/mm) k Non-Dimensional Initial Amplitude ka Atwood Number (Air-SF6) A 0.68 Shock Wave Mach Number M shock 1.19 Piston Velocity V piston Initial Growth Rate(mm/ms) a Function Generator Frequency f 3.24 Growth Reduction Factor ψ Table 6.1: Experimental parameters for spike (diagonal) experiment.

66 65 Spike (saddle) Parameter Symbol Value Initial Amplitude (mm) a Wave Number (1/mm) k Non-Dimensional Initial Amplitude ka Atwood Number (Air-SF6) A 0.68 Shock Wave Mach Number M shock 1.19 Piston Velocity V piston Initial Growth Rate(mm/ms) a Function Generator Frequency f 3.24 Growth Reduction Factor ψ Table 6.2: Experimental parameters for spike (saddle) experiment. Bubble (diagonal) Parameter Symbol Value Initial Amplitude (mm) a Wave Number (1/mm) k Non-Dimensional Initial Amplitude ka Atwood Number (Air-SF6) A 0.68 Shock Wave Mach Number M shock 1.19 Piston Velocity V piston Initial Growth Rate(mm/ms) a Function Generator Frequency f 3.24 Growth Reduction Factor ψ Table 6.3: Experimental parameters for bubble (diagonal) experiment

67 66 Bubble (saddle) Parameter Symbol Value Initial Amplitude (mm) a Wave Number (1/mm) k Non-Dimensional Initial Amplitude ka Atwood Number (Air-SF6) A 0.68 Shock Wave Mach Number M shock 1.19 Piston Velocity V piston Initial Growth Rate(mm/ms) a Function Generator Frequency f 3.24 Growth Reduction Factor ψ Table 6.4: Experimental parameters for bubble (saddle) experiment. One observation that can be made from the data in Tables 6.1 through 6.4 is that the initial amplitudes for the spike and the bubble are different. This might be the result of having only one speaker mounted at the bottom of test section. Having only a single speaker produces a pressure wave moving upward in the tube without a canceling pressure wave moving downward as occurs in experiments using two speakers such as in Ferguson s [10] three-dimensional random initial perturbation on gas-gas interface experiment. For each initial perturbation condition, seven experiments are run and recorded. However, only two experiments for each condition are presented. The rest of the experiments were not taken into consideration for the following reasons. For some experiments, the spike or bubble structure was not symmetric or not smooth due to imperfect initial perturbation. Also, occasionally it was too difficult to measure the amplitude accurately during the early time of some experiments because the tip of the spike structure was covered by or too close to the exit holes.

68 67 The image sequences in Figures 6.1 through 6.4 show the examples of the instability development of the four different initial perturbation conditions. Raw images and processed images are displayed side by side for each experimental time.

69 68 t = 0ms t = 0.335ms t = 0.835ms t = 1.335ms t = 1.835ms t = 2.335ms t = 2.835ms t = 3.335ms t = 3.835ms t = 4.335ms t = 4.835ms t = 5.335ms t = 5.835ms Flare t = 6.335ms flare Figure 6.1: Image sequence for a spike (diagonal) experiment. Figure 6.1 shows an image sequence for a typical spike (diagonal) experiment illustrating the growth of the instability with time. At the center of each image, a dark vertical line

70 69 occurs due to cutting off the area of corner and re-gluing the left and right parts of raw images. At later times of the experiment, the mushroom structure appears. Also, a flare structure appears along the interface. This structure has previously been observed by Long et al. [12] and occurs because two vortex tubes are intersecting with each other at those locations. t= 0ms t = ms t= 0.785ms t = ms t= 1.785ms t = ms t= 2.785ms t = ms t= 3.785ms t = ms t= 4.785ms t = ms t= 5.785ms t = ms Figure 6.2: Experimental amplitude growth for spike (saddle) experiment.

71 70 In Figure 6.2, although the amplitude of initial perturbation of spike (saddle) experiment increases with time, it does not grow as much as in the spike (diagonal) experiment. At the same time, the mushroom structure during the late time of the experiment is not as compact as the spike (diagonal) one. Furthermore, the flare structure does not appear in these images. t= 0ms t = ms t= 0.936ms t = ms t= 1.936ms t = ms t= 2.936ms t = ms t= 3.936ms t = ms t= 4.936ms t = ms t= 5.936ms t = ms Figure 6.3: Experimental amplitude growth for bubble (diagonal) experiment.

72 71 Similar to spike (diagonal) experiment, Figure 6.3 shows that the total amplitude of in the bubble (diagonal) experiments continuously grows with time. Also, both mushroom and flare structures appear in this sequence at late time in this experiment. t= 0.503ms t = ms t= 1.503ms t = ms t= 2.503ms t = ms t= 3.503ms t = ms t= 4.503ms t = ms t= 5.503ms t = ms t= Figure 6.4: Experimental amplitude growth for bubble (saddle) experiment. Figure 6.4 is an image sequence from a bubble (saddle) experiment showing that the amplitude grows slower than that of the bubble (diagonal) experiment. Also, during the later times of experiment, only mushroom structure appears and the flare structure is absent.

73 Results and Comparison to Linear Stability Theory Figure 6.5: Comparison of non-dimensional experimental amplitude measurements with linear stability theory [3] [4] showing all initial perturbation conditions. Figure 6.5 shows a comparison of measured amplitude versus time with that predicted by linear stability analysis. The spike amplitude agrees with the linear stability theory during early times while at late time the amplitude of spike becomes non-linear. For both the bubble (diagonal) and bubble (saddle) experiments the bubble amplitude also agrees with the linear instability during early times and becomes slightly non-linear later. But this is not as obvious as with the spike experiments. This might be because the initial amplitude of bubble experiments is significantly smaller than that of the spike experiments.

74 73 Figure 6.6: Comparison of experimental spike growth rate to linear stability theory in non-dimensional log-log scale. Figure 6.7: Comparison of experimental bubble growth rate to linear stability theory in non-dimensional log-log scale.

75 74 Figure 6.6 and 6.7 show a comparison of measured growth rate of both spikes and bubbles versus time with that predicted by linear stability analysis in log-log scale. Linear stability theory predicts the growth rate to remain constant, a a 0 = 1. In the experimental results for both the spike and the bubble the growth rate remains constant at early times, and then decreases, indicating that a non-linear model should be used for comparison at the later times Results and Comparison to Zhang-Sohn Model Figure 6.8: Comparison of experimental amplitude measurement to the Zhang & Sohn model [20] [21] with all initial perturbation condition in non-dimensional scale. Figure 6.8 shows a comparison of measured amplitude versus time with that predicted by the Zhang & Sohn Model [20] [21]. The experimental results seem not to agree with it very well. For spike, the model spike amplitude grows faster during the

76 75 early time of experiment. The model curve is higher than that of experiment. During the late time, two spike amplitude curves seem to be parallel to each other. For bubble, the model bubble amplitude grows slower during the whole experiment. The model curve is lower than that of experiment. Figure 6.9: Comparison of experimental spike growth rate to Zhang &Sohn model [20] [21] in non-dimensional log-log scale.

77 76 Figure 6.10: Comparison of experimental bubble growth rate to Zhang &Sohn model [20] [21] in non-dimensional log-log scale. Figure 6.9 and 6.10 show a comparison of measured growth rate of spike/bubble versus time with that predicted by Zhang & Sohn model [20] [21] in log-log scale. The spike growth rate of model is higher than that of the experiment. Later, it drops quickly and matches some experimental data points during the late time. The bubble growth rate of model drops faster during the early time of the experiment. And later, it tends to be constant, and matches some experimental data points.

78 Results and Comparison to Modified 3D Sadot et al. Model Figure 6.11: Comparison of experimental amplitude measurement to modified Sadot et al. model [22] with all initial perturbation condition in non-dimensional scale. Figure 6.11 shows a comparison of measured amplitude versus time with that predicted by Modified 3D Sadot et al. Model [22]. The experimental results match modified Sadot et al. model better than Zhang &Sohn model [20] [21]. However, there are still some differences between them. For spike, the model spike amplitude does match some experimental results at the beginning the experiment. Later, the model spike amplitude becomes smaller. And the model curve is lower than that of experiment. For bubble, the model bubble amplitude grows faster during the early time and concaves down during the late time.

79 78 Figure 6.12: Comparison of experimental spike growth rate to modified Sadot et al. model [22] in non-dimensional log-log scale. Figure 6.13: Comparison of experimental bubble growth rate to modified Sadot et al. model [22] in non-dimensional log-log scale.

80 79 Figure 6.12 and 6.13 show a comparison of measured growth rate of spike/bubble versus time with that predicted by modified 3D Sadot et al. model [22] in log-log scale. The model spike growth rate drops faster than that of experiment. During the late time, it matches some experimental results. It can explain why the model spike curve is parallel to some of experimental curve during the late time. The growth rate of bubble is higher than that of experimental results and later drops into the experimental data range Results and Comparison to Late-Time (Goncharov) Model Figure 6.14: Comparison of experimental spike growth rate to late-time (Goncharov) model [25] in non-dimensional log-log scale.

81 80 Figure 6.15: Comparison of experimental bubble growth rate to late-time (Goncharov) model [25] in non-dimensional log-log scale. Figure 6.14 and 6.15 show a comparison of measured growth rate of spike/bubble versus time with that predicted by Late Time (Goncharov) model [25] in log-log scale. At late time this model predicts that the growth rate should be a straight line with a slope of -1 in a log-log plot. While the experimental measurements do seem to lie on a straight line at late time, the slope of this line differs significantly from that predicted by the model. The experimental slope of spike experiments is and that of the bubble experiments is This disagreement may be the result of the experiments not going out to late enough time to observe the true asymptotic result.

82 Circulation Using the DaVis program from LaVision, the vorticity field of each experiment was calculated by PIV processing. The results are shown in figures 6.16 and t = 0.835ms t = 1.335ms t = 1.835ms t = 2.335ms t = 2.835ms t = 3.335ms t = 3.835ms t = 4.335ms t = 4.835ms t = 5.335ms t = 5.835ms t = 6.335ms Figure 6.16: The vorticity evolution for a spike (diagonal) experiment.

83 82 Figure 6.16 shows the vorticity evolution for a typical spike (diagonal) experiment. As shown in images, the vorticity increases in area and keeps the same magnitude in strength. At the same time, it is not only concentrated in the mushroom region, but also exists along the interface, where the flare structure appears. t= 0.785ms t = ms t= 1.785ms t = ms t= 3.285ms t = ms t= 4.285ms t = ms t= 5.785ms t = ms Figure 6.17: The vorticity evolution for spike (saddle) experiment. Figure 6.17 shows the vorticity evolution for a typical spike (saddle) experiment. Similarly to as observed in figure 6.16, the vorticity field increases in area, keeps

84 83 constant in magnitude and exists along the interface. However, when compared to the results of the spike (diagonal) experiment, the area of vorticity field is smaller. However, magnitude of vorticity is greater, which shows a deeper color. Also, the vortex cores of the spike (saddle) experiment is not as compact as that of spike (diagonal) experiment, appearing more similar to a 2D vortex. Based on the three different methods described in section 5.3.4, the results of circulation calculation are shown in figures 6.18 through Figure 6.18 shows circulation vs. time for spike (diagonal) experiment 1. Figure 6.19 shows circulation vs. time for spike (diagonal) experiment 2. Figure 6.20 shows circulation vs. time for spike (saddle) experiment 1. Figure 6.21shows circulation vs. time for spike (saddle) experiment 2. Figure 6.18: Circulation vs. Time for spike (diagonal) 1 experiment.

85 84 Figure 6.19: Circulation vs. Time for spike (diagonal) 2 experiment. Figure 6.20: Circulation vs. Time for spike (saddle) 1 experiment.

86 85 Figure 6.21: Circulation vs. Time for spike (saddle) 2 experiment. As shown in each of these figures, the circulations calculated using each of the three methods agree well with each other. So, in the results that follow the circulation measurement is set to be the average value of three methods. It is useful to compare the circulation measurements with those predicted by a simple incompressible model for RMI. In the 2D case, if the initial perturbation is very small, the linear stability theory can be applied to give an estimation of the circulation. For a uniformly spaced vortex row, the stream-function can be estimated by Jacobs and Sheeley [27] as, ψ = Г (kx) ln (cosh(kz)+cos ) (6.1) 4π cosh(kz) cos (kx)

87 86 A 3D periodic perturbation on the interface is η = a(t) cos(k x x) cos(k y y) (6.2) The velocity potentials in the two fluids are Φ 1 = b(t)e kz cos (k x x)cos(k y y) (6.3) Φ 2 = b(t)e kz cos (k x x)cos(k y y) (6.4) where k = k x 2 + k y 2 (k x = k y ) (6.5) a = kb (6.6) If the perturbation amplitude is sufficiently small, the method of linear stability theory can be applied to calculate the initial vorticity distribution. The strength of vortex sheet (γ) is γ x = Φ 2 x z=0 Φ 1 x z=0 (6.7) = x ( b(t)e kz cos(k x x)cos(k y y)) z=0 x (b(t)ekz cos (k x x)cos(k y y)) z=0 = b(t)( k x sin(k x x)) cos(k y y) b(t)(( k x sin(k x x)) cos(k y y) = 2b(t)k x sin(k x x) cos(k y y) γ y = Φ 2 Φ 1 y z=0 y z=0 (6.8)

88 87 = 2b(t)k y sin(k y y) cos(k x x) Figure 6.22: Two coordinate systems, x-y for spike (diagonal) structure and x -y for spike (saddle) structure. For the spike (diagonal) experiment, it can be assumed that the initial perturbation is located right at the center of the shock tube and the laser sheet is on the x z plane where y=0 γ x = 2b(t)k x sin(k x x) (6.9) γ y = 0 (6.10) The circulation for this case is Γ = π kx 0 γdx (6.12) π kx = 2b(t)k x sin(k x x) dx 0 = 2b(t)( cos(k x x)) 0 π kx = 4b(t) = 4a k